Jones polynomial

[1][2] Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable

This results in a Laurent polynomial with integer coefficients in the variable

This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link.

The construction was developed by Vladimir Turaev and published in 1990.

denote the set of all isotopic types of tangle diagrams, with

ends, having no crossing points and no closed components (smoothings).

is the ring of Laurent polynomials with integer coefficients in the variable

Jones' original formulation of his polynomial came from his study of operator algebras.

In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. the Potts model, in statistical mechanics.

A theorem of Alexander states that it is the trace closure of a braid, say with n strands.

of the braid group on n strands, Bn, into the Temperley–Lieb algebra

The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein relation: where

are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below: The definition of the Jones polynomial by the bracket makes it simple to show that for a knot

There exist an infinite number of non-equivalent knots that have the same Jones polynomial.

An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.

In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) of

One thinks of the strands of a link as being "colored" by a representation, hence the name.

, colored Jones polynomials satisfy the following two properties:[8] These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of

can be given a combinatorial description using the Jones-Wenzl idempotents, as follows: The resulting element of

As first shown by Edward Witten,[10] the Jones polynomial of a given knot

, and computing the vacuum expectation value of a Wilson loop

of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be the Vassiliev invariant of the knot

In order to unify the Vassiliev invariants (or, finite type invariants), Maxim Kontsevich constructed the Kontsevich integral.

The value of the Kontsevich integral, which is the infinite sum of 1, 3-valued chord diagrams, named the Jacobi chord diagrams, reproduces the Jones polynomial along with the

By numerical examinations on some hyperbolic knots, Rinat Kashaev discovered that substituting the n-th root of unity into the parameter of the colored Jones polynomial corresponding to the n-dimensional representation, and limiting it as n grows to infinity, the limit value would give the hyperbolic volume of the knot complement.

In 2000 Mikhail Khovanov constructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (see Khovanov homology).

It is an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot.

It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite.

[11] It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.